LTI 系统的响应

有这么几种响应的类型：

脉冲响应（impulse response）

和系统的传递函数相关
阶跃响应（step response）

在实践中比较常见，与一些 LTI 系统中的物理参数有关
正弦响应（sinusoidal response）

和系统的频率响应相关

定义

A rational transfer function can be described either as a ratio of two polynomials in \(s\)

\[
H(s) = \frac{b(s)}{a(s)} = \frac{b_ms^m + b_{m-1}s^{m-1}+\cdots + b_1s + b_0}{a_ns^n + a_{n-1}s^{n-1}+\cdots + a_1s + a_0}
\]

or as a ratio in factored pole-zero form

\[
H(s) = K \frac{\prod_{i=1}^m (s - z_i)}{\prod_{i-1}^n (s - p_i)}
\]

\(K = b_m / a_n\)
the roots of the numerator, \(z_1, z_2, \ldots , z_m\) are called the finite zeros of the system
the roots of the denominator, \(p_1, p_2, \ldots , p_n\) are called the finite poles of the system
assuming the coefficients of \(a\) and \(b\) are real, complex poles or zeros come in complex conjugate pairs

Poles and zeros plots

Poles and zeros of a rational functions are often shown in a pole-zero plot: poles marked by “×”, zeros marked by “○”.

例子：

\[
F(s) = k \frac{(s+1.5)(s^2+2s+5)}{(s+2.5)(s-2)(s^2-2s+2)}
\]

脉冲响应

定义

一个 LTI 系统的脉冲响应是系统在输入信号\(u(t)\)为 unit impulse function 也就是\(\delta (t)\)且伴随 zero inital conditions 时的输出信号。

Suppose the transfer function representation of a LTI system is

\[
G(s) = \frac{Y(s)}{U(s)}
\]

Its impulse response is given by \(u(t) = \delta (t) \rightarrow U(s) = 1\). We then have

\[
Y(s) = G(s)
\]

Inverse LT gives

\[
y(t) = \mathcal{L}^{-1}\{ G(s) \} = g(t)
\]

First-order system

考虑一个 first-order system

\[
G(s) = \frac{1}{s + \sigma}
\]

使用拉普拉斯逆变换可得

\[
g(t) = e^{-\sigma t}
\]

可以发现当\(\sigma > 0\)时，随着\(t \rightarrow \infty\)，\(g(t) \rightarrow 0\)。那么脉冲响应是稳定（stable）的。

如果\(\sigma < 0\)，指数项随着时间增长，脉冲响应不稳定（unstable）。

\(\sigma\)越大，衰减到零的速度越快。

Second-order system

一个标准的 2nd order transfer function 可被表示为

\[
H(s) = \frac{K \omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} = \frac{K\omega_n^2}{(s + \zeta\omega_n)^2 + \omega_n^2(1 - \zeta^2)}
\]

其中\(K\)是系统的 Gain。

A pair of complex poles, can be defined in terms of their real and imaginary parts, as follows:

\[
s = -\sigma \pm j\omega_d
\]

where

\[
\sigma = \zeta \omega_n \ \mbox{and}\ \omega_d = \omega_n \sqrt{1 - \zeta^2}
\]

\(\zeta\) is the damping ratio and \(\omega_n\) is the undamped natural frequency.

它的脉冲响应为

\[
y(t) = h(t) = \frac{K\omega_n}{\sqrt{1 - \zeta^2}}e^{-\sigma t}\sin (\omega_d t)
\]

脉冲响应和 s-plane 上 pole locations 的关系

real, positive poles correspond to growing exponential terms
real, negative poles correspond to decaying exponential terms
a pole at \(s = 0\) corresponds to a constant term
complex pole pairs with positive real part correspond to exponentially growing sinusoidal terms
complex pole pairs with negative real part correspond to exponentially decaying sinusoidal terms
pure imaginary pole pairs correspond to sinusoidal terms
repeated poles yield same types of terms, multiplied by powers of \(t\)

阶跃响应

定义

The step response is the response of the system \(G(s)\) to a unit step input \(u(t) = t\).

因为\(U(s) = \frac{1}{s}\)，所以可得阶跃响应

\[
y(t) = \mathcal{L}^{-1}\left\{\frac{G(s)}{s}\right\}
\]

DC gain

Static/Steady-state/DC gain: this is the ratio of the output of a system to its input (presumed constant) after all transients have decayed.

If the magnitude of the step input is \(A\), i.e. \(u(t) = AU(t) \rightarrow U(s) = \frac{A}{s}\), we then have (via Final Value Theorem)

\[
\lim_{t \rightarrow \infty} y(t) = \lim_{s \rightarrow 0}sY(s) = \lim_{s \rightarrow 0}sG(s)\frac{A}{s} = AG(0)
\]

Hence, DC gain,

\[
K = \frac{\lim_{t \rightarrow \infty}y(t)}{\lim_{t \rightarrow \infty}u(t)} = \frac{AG(0)}{A} = G(0)
\]

Integrator

\[
y(t) = \int_0^t K_iu(\tau)\mathrm{d}\tau
\]

其中\(K_i\)是 integrator gain。

传递函数为

\[
G(s) = \frac{Y(s)}{U(s)} = \frac{K_i}{s}
\]

阶跃响应为

\[
Y(s) = G(s)U(s) = G(s)\frac{1}{s} = \frac{K_i}{s}
\]

通过拉普拉斯逆变换得

\[
y(t) = K_i t
\]

Differentiator

\[
y(t) = K_d \frac{\mathrm{d}u(t)}{\mathrm{d}t}
\]

其中\(K_d\)是 derivative gain。

传递函数为

\[
G(s) = \frac{Y(s)}{U(s)} = K_d s
\]

阶跃响应为

\[
Y(s) = G(s)U(s) = \frac{G(s)}{s} = K_d
\]

通过拉普拉斯逆变换得

\[
y(t) = K_d \delta (t)
\]

First-order systems

\[
\tau \frac{\mathrm{d}y(t)}{\mathrm{d}t} + y(t) = Ku(t)
\]

其中\(K\)是 steady-state/static gain 并且\(\tau\)是时间常数（time constant）。

传递函数为

\[
G(s) = \frac{Y(s)}{U(s)} = \frac{K}{\tau s + 1};\ y(0) = 0
\]

阶跃响应为

\[
Y(s) = G(s)U(s) = \frac{K}{\tau s + 1}\frac{1}{s} = \frac{K}{s} - \frac{K\tau}{\tau s + 1}
\]

通过拉普拉斯逆变换得

\[
y(t) = K - Ke^{-t/\tau}
\]

Second-order systems

\[
\frac{\mathrm{d}^2 y(t)}{\mathrm{d} t^2} + 2\zeta \omega_n \frac{\mathrm{d}y(t)}{\mathrm{d}t} + \omega_n^2y(t) = K\omega_n^2u(t)
\]

其中\(K\)，\(\zeta\)和\(\omega_n\)分别是 steady-state/static gain，damping ratio 和 undamped natural frequency。

传递函数为

\[
G(s) = \frac{Y(s)}{U(s)} = \frac{K\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2},\ y(0) = y\prime (0) = 0
\]

阶跃响应为

\[
Y(s) = \ldots = \frac{K}{s} - \frac{K(s + \zeta\omega_n)}{(s + \zeta\omega_n)^2 + \omega_n^2(1-\zeta^2)} -\frac{K\zeta}{\sqrt{1-\zeta^2}}\frac{\omega_n\sqrt{1-\zeta^2}}{(s + \zeta\omega_n)^2 + \omega_n^2(1 - \zeta^2)}
\]

通过拉普拉斯逆变换得

\[
y(t) = \ldots = K\left(1 - \frac{e^{-\sigma t}}{\sqrt{1-\zeta^2}}\sin [\omega_d t + \phi]\right)
\]

\(|\mathcal{R}\{s\}| = \sigma = \zeta \omega_{n}\)决定 exponential envelope。

\(\mathcal{L}\{s\} = \omega_{d} = \omega_{n}\sqrt{1 - \zeta^2}\)决定 frequency of the sinusoidal signal。

阶跃响应和 s-plane 上 pole locations 的关系

\(K\) determines the steady state output response to a step input.
3 types of responses possible: underdamped \(\zeta < 1\), overdamped \(\zeta > 1\) and critically damped \(\zeta = 1\).
For underdamped response
- real part of pole (\(\sigma = \zeta \omega_n\)) determines how quickly the oscillations decay away
- imaginary part of pole (\(\omega_d = \omega_n\sqrt{1 - \zeta^2}\)) gives you the frequency of the oscillation
For over- and critically damped systems, they behave more like first order systems, except more sluggish.

示例：https://lpsa.swarthmore.edu/SecondOrder/SOI.html

时域上的一些规范

在控制系统设计中，以下几个时域上的规范经常被用到：

rise time, \(t_r\): the time it takes the system to reach the vicinity of its new set point
settling time, \(t_s\): the time it takes the system transient to decay
overshoot, \(M_p\): the maximum amount the system overshoots its final value divided by its final value
peak time, \(t_p\): the time it takes the system to reach the maximum overshoot point

Rise time

最常见的定义为系统响应从10%到90%稳态值所需要的时间。

不太容易能得到准确的表达式。

对于一个标准的 2nd-order 传递函数，可得近似：

\[
t_r = \frac{2.16\zeta + 0.60}{\omega_n}\ \mbox{for}\ 0.3 \leq \zeta \leq 0.8
\]

取平均值\(\zeta = 0.55\)得

\[
t \approx \frac{1.8}{\omega_n}
\]

注意：仅适用无 zero 的 2nd-order system，其它系统会有较大误差。

对于给定的\(\zeta\)，越大的\(\omega_n\)会有更快的响应：

对于给定的\(\omega_n\)，更小的\(\zeta\)会让响应稍微加快：

Overshoot

\[
M_p = Ke^{-\pi \zeta / \sqrt{1 - \zeta^2}},\ 0\leq \zeta <1
\]

百分比形式：

\[
\%M_p = e^{-\pi\zeta / \sqrt{1 - \zeta^2}} \times 100\%
\]

Settling time

This is the time required for the transient to decay to a small value so that \(y(t)\) is almost in the steady-state.

Measure of smallness: 1%; 2% or 5% have been used.

Notice that the deviation of \(y\) from \(K\) is enclosed by the envelop of the exponential function

\[
K\left(1 \pm \frac{e^{-\zeta \omega_n t}}{\sqrt{1 - \zeta^2}}\right)
\]

那么计算2%的 settling time：

\[
\begin{align}
\frac{e^{-\zeta \omega_n t}}{\sqrt{1 - \zeta^2}} &= 0.02
\newline
e^{-\zeta\omega_n t} &\approx 0.02
\newline
\zeta \omega_n t_s &\approx 4
\newline
t_s &= \frac{4}{\zeta\omega_n} = \frac{4}{\sigma}
\end{align}
\]

Design Synthesis

Selection of pole and zero locations to meet these time-domain specifications for dynamic response.

稳定性

A linear time-invariant system is said to be stable if all the roots of the transfer function denominator polynomial have negative real-parts (i.e. all in the left half plane, \(\sigma < 0\)) and is unstable otherwise (\(\sigma > 0\)).

The real-part of the pole determine its stability; for a stable system, all poles must be in the LHP.

If the system has any poles in the RHP, it is unstable.

If the system has nonrepeated \(j\omega\)-axis poles, it is marginally stable.